3.990 \(\int \frac{1}{(c x)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{256 \left (a+b x^2\right )^{11/4}}{77 a^4 c (c x)^{11/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{7 a^3 c (c x)^{11/2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}+\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}} \]

[Out]

2/(a*c*(c*x)^(11/2)*(a + b*x^2)^(1/4)) - (8*(a + b*x^2)^(3/4))/(a^2*c*(c*x)^(11/
2)) + (64*(a + b*x^2)^(7/4))/(7*a^3*c*(c*x)^(11/2)) - (256*(a + b*x^2)^(11/4))/(
77*a^4*c*(c*x)^(11/2))

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Rubi [A]  time = 0.126085, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{256 \left (a+b x^2\right )^{11/4}}{77 a^4 c (c x)^{11/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{7 a^3 c (c x)^{11/2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}+\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(13/2)*(a + b*x^2)^(5/4)),x]

[Out]

2/(a*c*(c*x)^(11/2)*(a + b*x^2)^(1/4)) - (8*(a + b*x^2)^(3/4))/(a^2*c*(c*x)^(11/
2)) + (64*(a + b*x^2)^(7/4))/(7*a^3*c*(c*x)^(11/2)) - (256*(a + b*x^2)^(11/4))/(
77*a^4*c*(c*x)^(11/2))

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Rubi in Sympy [A]  time = 15.7196, size = 95, normalized size = 0.87 \[ \frac{2}{a c \left (c x\right )^{\frac{11}{2}} \sqrt [4]{a + b x^{2}}} - \frac{8 \left (a + b x^{2}\right )^{\frac{3}{4}}}{a^{2} c \left (c x\right )^{\frac{11}{2}}} + \frac{64 \left (a + b x^{2}\right )^{\frac{7}{4}}}{7 a^{3} c \left (c x\right )^{\frac{11}{2}}} - \frac{256 \left (a + b x^{2}\right )^{\frac{11}{4}}}{77 a^{4} c \left (c x\right )^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(13/2)/(b*x**2+a)**(5/4),x)

[Out]

2/(a*c*(c*x)**(11/2)*(a + b*x**2)**(1/4)) - 8*(a + b*x**2)**(3/4)/(a**2*c*(c*x)*
*(11/2)) + 64*(a + b*x**2)**(7/4)/(7*a**3*c*(c*x)**(11/2)) - 256*(a + b*x**2)**(
11/4)/(77*a**4*c*(c*x)**(11/2))

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Mathematica [A]  time = 0.0624786, size = 63, normalized size = 0.58 \[ -\frac{2 \sqrt{c x} \left (7 a^3-12 a^2 b x^2+32 a b^2 x^4+128 b^3 x^6\right )}{77 a^4 c^7 x^6 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(13/2)*(a + b*x^2)^(5/4)),x]

[Out]

(-2*Sqrt[c*x]*(7*a^3 - 12*a^2*b*x^2 + 32*a*b^2*x^4 + 128*b^3*x^6))/(77*a^4*c^7*x
^6*(a + b*x^2)^(1/4))

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Maple [A]  time = 0.008, size = 53, normalized size = 0.5 \[ -{\frac{2\,x \left ( 128\,{b}^{3}{x}^{6}+32\,a{b}^{2}{x}^{4}-12\,{a}^{2}b{x}^{2}+7\,{a}^{3} \right ) }{77\,{a}^{4}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( cx \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(13/2)/(b*x^2+a)^(5/4),x)

[Out]

-2/77*x*(128*b^3*x^6+32*a*b^2*x^4-12*a^2*b*x^2+7*a^3)/(b*x^2+a)^(1/4)/a^4/(c*x)^
(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(13/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(13/2)), x)

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Fricas [A]  time = 0.20905, size = 77, normalized size = 0.71 \[ -\frac{2 \,{\left (128 \, b^{3} x^{6} + 32 \, a b^{2} x^{4} - 12 \, a^{2} b x^{2} + 7 \, a^{3}\right )}}{77 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} a^{4} c^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(13/2)),x, algorithm="fricas")

[Out]

-2/77*(128*b^3*x^6 + 32*a*b^2*x^4 - 12*a^2*b*x^2 + 7*a^3)/((b*x^2 + a)^(1/4)*sqr
t(c*x)*a^4*c^6*x^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x)**(13/2)/(b*x**2+a)**(5/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(13/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(13/2)), x)